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Register a new userComplex dimensions of fractals and meromorphic extensions of fractal zeta functions
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We study meromorphic extensions of distance and tube zeta functions, as well as of geometric zeta functions of fractal strings. The distance zeta function $zeta_A(s):=int_{A_delta} d(x,A)^{s-N}mathrm{d}x$, where $delta>0$ is fixed and $d(x,A)$ denotes the Euclidean distance from $x$ to $A$ extends the definition of the zeta function associated with bounded fractal strings to arbitrary bounded subsets $A$ of $mathbb{R}^N$. The abscissa of Lebesgue convergence $D(zeta_A)$ coincides with $D:=overlinedim_BA$, the upper box dimension of $A$. The complex dimensions of $A$ are the poles of the meromorphic continuation of the fractal zeta function of $A$ to a suitable connected neighborhood of the critical line ${Re(s)=D}$. We establish several meromorphic extension results, assuming some suitable information about the second term of the asymptotic expansion of the tube function $|A_t|$ as $tto0^+$, where $A_t$ is the Euclidean $t$-neighborhood of $A$. We pay particular attention to a class of Minkowski measurable sets, such that $|A_t|=t^{N-D}(mathcal M+O(t^gamma))$ as $tto0^+$, with $gamma>0$, and to a class of Minkowski nonmeasurable sets, such that $|A_t|=t^{N-D}(G(log t^{-1})+O(t^gamma))$ as $tto0^+$, where $G$ is a nonconstant periodic function and $gamma>0$. In both cases, we show that $zeta_A$ can be meromorphically extended (at least) to the open right half-plane ${Re(s)>D-gamma}$. Furthermore, up to a multiplicative constant, the residue of $zeta_A$ evaluated at $s=D$ is shown to be equal to $mathcal M$ (the Minkowski content of $A$) and to the mean value of $G$ (the average Minkowski content of $A$), respectively. Moreover, we construct a class of fractal strings with principal complex dimensions of any prescribed order, as well as with an infinite number of essential singularities on the critical line ${Re(s)=D}$.
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The theory of zeta functions of fractal strings has been initiated by the first author in the early 1990s, and developed jointly with his collaborators during almost two decades of intensive research in numerous articles and several monographs. In 2009, the same author introduced a new class of zeta functions, called `distance zeta functions, which since then, has enabled us to extend the existing theory of zeta functions of fractal strings and sprays to arbitrary bounded (fractal) sets in Euclidean spaces of any dimension. A natural and closely related tool for the study of distance zeta functions is the class of tube zeta functions, defined using the tube function of a fractal set. These three classes of zeta functions, under the name of fractal zeta functions, exhibit deep connections with Minkowski contents and upper box dimensions, as well as, more generally, with the complex dimensions of fractal sets. Further extensions include zeta functions of relative fractal drums, the box dimension of which can assume negative values, including minus infinity. We also survey some results concerning the existence of the meromorphic extensions of the spectral zeta functions of fractal drums, based in an essential way on earlier results of the first author on the spectral (or eigenvalue) asymptotics of fractal drums. It follows from these results that the associated spectral zeta function has a (nontrivial) meromorphic extension, and we use some of our results about fractal zeta functions to show the new fact according to which the upper bound obtained for the corresponding abscissa of meromorphic convergence is optimal. Finally, we conclude this survey article by proposing several open problems and directions for future research in this area.
We study the essential singularities of geometric zeta functions $zeta_{mathcal L}$, associated with bounded fractal strings $mathcal L$. For any three prescribed real numbers $D_{infty}$, $D_1$ and $D$ in $[0,1]$, such that $D_{infty}<D_1le D$, we construct a bounded fractal string $mathcal L$ such that $D_{rm par}(zeta_{mathcal L})=D_{infty}$, $D_{rm mer}(zeta_{mathcal L})=D_1$ and $D(zeta_{mathcal L})=D$. Here, $D(zeta_{mathcal L})$ is the abscissa of absolute convergence of $zeta_{mathcal L}$, $D_{rm mer}(zeta_{mathcal L})$ is the abscissa of meromorphic continuation of $zeta_{mathcal L}$, while $D_{rm par}(zeta_{mathcal L})$ is the infimum of all positive real numbers $alpha$ such that $zeta_{mathcal L}$ is holomorphic in the open right half-plane ${{rm Re}, s>alpha}$, except for possible isolated singularities in this half-plane. Defining $mathcal L$ as the disjoint union of a sequence of suitable generalized Cantor strings, we show that the set of accumulation points of the set $S_{infty}$ of essential singularities of $zeta_{mathcal L}$, contained in the open right half-plane ${{rm Re}, s>D_{infty}}$, coincides with the vertical line ${{rm Re}, s=D_{infty}}$. We extend this construction to the case of distance zeta functions $zeta_A$ of compact sets $A$ in $mathbb{R}^N$, for any positive integer $N$.
In 2009, the first author introduced a new class of zeta functions, called `distance zeta functions, associated with arbitrary compact fractal subsets of Euclidean spaces of arbitrary dimension. It represents a natural, but nontrivial extension of the theory of `geometric zeta functions of bounded fractal strings. In this memoir, we introduce the class of `relative fractal drums (or RFDs), which contains the classes of bounded fractal strings and of compact fractal subsets of Euclidean spaces as special cases. Furthermore, the associated (relative) distance zeta functions of RFDs, extend (in a suitable sense) the aforementioned classes of fractal zeta functions. This notion is very general and flexible, enabling us to view practically all of the previously studied aspects of the theory of fractal zeta functions from a unified perspective as well as to go well beyond the previous theory. The abscissa of (absolute) convergence of any relative fractal drum is equal to the relative box dimension of the RFD. We pay particular attention to the question of constructing meromorphic extensions of the distance zeta functions of RFDs, as well as to the construction of transcendentally $infty$-quasiperiodic RFDs (i.e., roughly, RFDs with infinitely many quasiperiods, all of which are algebraically independent). We also describe a class of RFDs (and, in particular, a new class of bounded sets), called {em maximal hyperfractals}, such that the critical line of (absolute) convergence consists solely of nonremovable singularities of the associated relative distance zeta functions. Finally, we also describe a class of Minkowski measurable RFDs which possess an infinite sequence of complex dimensions of arbitrary multiplicity $mge1$, and even an infinite sequence of essential singularities along the critical line.
Recently, the first author has extended the definition of the zeta function associated with fractal strings to arbitrary bounded subsets $A$ of the $N$-dimensional Euclidean space ${mathbb R}^N$, for any integer $Nge1$. It is defined by $zeta_A(s)=int_{A_{delta}}d(x,A)^{s-N},mathrm{d} x$ for all $sinmathbb{C}$ with $operatorname{Re},s$ sufficiently large, and we call it the distance zeta function of $A$. Here, $d(x,A)$ denotes the Euclidean distance from $x$ to $A$ and $A_{delta}$ is the $delta$-neighborhood of $A$, where $delta$ is a fixed positive real number. We prove that the abscissa of absolute convergence of $zeta_A$ is equal to $overlinedim_BA$, the upper box (or Minkowski) dimension of $A$. Particular attention is payed to the principal complex dimensions of $A$, defined as the set of poles of $zeta_A$ located on the critical line ${mathop{mathrm{Re}} s=overlinedim_BA}$, provided $zeta_A$ possesses a meromorphic extension to a neighborhood of the critical line. We also introduce a new, closely related zeta function, $tildezeta_A(s)=int_0^{delta} t^{s-N-1}|A_t|,mathrm{d} t$, called the tube zeta function of $A$. Assuming that $A$ is Minkowski measurable, we show that, under some mild conditions, the residue of $tildezeta_A$ computed at $D=dim_BA$ (the box dimension of $A$), is equal to the Minkowski content of $A$. More generally, without assuming that $A$ is Minkowski measurable, we show that the residue is squeezed between the lower and upper Minkowski contents of $A$. We also introduce transcendentally quasiperiodic sets, and construct a class of such sets, using generalized Cantor sets, along with Bakers theorem from the theory of transcendental numbers.
In 2009, the first author introduced a class of zeta functions, called `distance zeta functions, which has enabled us to extend the existing theory of zeta functions of fractal strings and sprays (initiated by the first author and his collaborators in the early 1990s) to arbitrary bounded (fractal) sets in Euclidean spaces of any dimensions. A closely related tool is the class of `tube zeta functions, defined using the tube function of a fractal set. These zeta functions exhibit deep connections with Minkowski contents and upper box (or Minkowski) dimensions, as well as, more generally, with the complex dimensions of fractal sets. In particular, the abscissa of (Lebesgue, i.e., absolute) convergence of the distance zeta function coincides with the upper box dimension of a set. We also introduce a class of transcendentally quasiperiodic sets, and describe their construction based on a sequence of carefully chosen generalized Cantor sets with two auxilliary parameters. As a result, we obtain a family of maximally hyperfractal compact sets and relative fractal drums (i.e., such that the associated fractal zeta functions have a singularity at every point of the critical line of convergence). Finally, we discuss the general fractal tube formulas and the Minkowski measurability criterion obtained by the authors in the context of relative fractal drums (and, in particular, of bounded subsets of the N-dimensional Euclidean space).
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